Viscosity and Terminal Velocity

Viscosity measures a fluid’s internal resistance to flow — how strongly adjacent layers drag on each other. It is quantified by the $\text{coefficient of viscosity}$ $\eta$, measured in $\mathrm{Pa}\mathrm{s}$.

In $\text{laminar}$ flow the fluid moves in smooth parallel layers, with no mixing between them and velocity at any point constant over time. In $\text{turbulent}$ flow the motion becomes chaotic, with eddies and swirls, and the drag rises sharply.

Viscosity is strongly temperature dependent. Most liquids become much less viscous when warmed — which is why engine oil flows more freely once warm, and why temperature must be controlled in the viscosity practical.

For a small sphere moving slowly through a fluid in laminar flow, the viscous drag is $F=6\pi \eta rv$, where $r$ is the sphere’s radius and $v$ its speed relative to the fluid.

The drag is proportional to speed. That linearity is what makes terminal velocity so easy to reach and to analyse: as the sphere speeds up, the drag grows in step until it balances the other forces.

The law only holds for small spheres, low speeds and laminar flow. It fails for large or fast objects, where turbulence makes drag roughly proportional to $v^2$ instead.

A sphere released in a liquid has three forces on it: weight downwards, upthrust upwards, and viscous drag upwards (opposing its downward motion).

At release the speed is zero, so drag is zero and the resultant force is weight minus upthrust — the sphere accelerates. As it speeds up, drag increases, the resultant falls, and so does the acceleration.

Eventually drag grows until weight equals upthrust plus drag. The resultant force is then zero, and by Newton’s first law the sphere continues at constant speed: its $\text{terminal velocity}$.

Substituting each force for a sphere of density ${\rho }_s$ in a fluid of density ${\rho }_f$: weight $\frac{4}{3}\pi r^3{\rho }_{s}g$, upthrust $\frac{4}{3}\pi r^3{\rho }_{f}g$, drag $6\pi \eta r{v}_t$.

Setting weight equal to upthrust plus drag and rearranging gives the terminal velocity. It grows with $r^2$ — so doubling the radius quadruples the terminal speed — and falls with increasing viscosity.

In the practical, measuring $v_t$ with light gates or by timing between markers lets you calculate $\eta$. Markers are placed far enough down the tube that the ball has already reached terminal velocity before timing starts.